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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Digital Gnosis

Frege’s philosophy of mathematics—Interview with Nathan Ormond, December 2021

Posted on October 10, 2021 by Joel David Hamkins
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I was interviewed by Nathan Ormond for a discussion on Frege’s philosophy of mathematics for his YouTube channel, Digital Gnosis, on 10 December 2021 at 4pm.

The interview concludes with a public comment and question & answer session.

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Posted in Events, Talks, Videos | Tagged Digital Gnosis, Frege, philosophy of mathematics | Leave a reply

Infinitely More

Abstraction in the function concept

A century ago mathematics witnessed a dramatic enlargement and abstraction of this central concept. Let's explore some of the mind-expanding new possibilities...

Joel David Hamkins
Jun 9
9
On going first

Would you rather go first or second? In many games, there is a definite advantage one way or the other. How can we redress these imbalances, if we seek to make truly fair and balanced games?

Joel David Hamkins
May 30
5
How we might have viewed the continuum hypothesis as a fundamental axiom necessary for mathematics

By mounting a philosophical historical thought experiment, I argue that our attitude toward the continuum hypothesis could easily have been very different than it is.

Joel David Hamkins
May 22
8
12

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Proof and the Art of Mathematics, MIT Press, 2020

Recent Comments

  • Joel David Hamkins on Lectures on Set Theory, Beijing, June 2025
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  • Mohammad Golshani on Lectures on Set Theory, Beijing, June 2025
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  • Joel David Hamkins on Skolem’s paradox and the countable transitive submodel theorem, Leeds Set Theory Seminar, May 2025

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  • Comment by Joel David Hamkins on A new and subtle order-theoretic fixed point theorem
    Your remarks about transfinite recursion strike me as polemical (and I believe this is the main reason for the downvotes). You make it sound cumbersome and complicated, perhaps even untrustworthy, whereas I know it as elegant and simple, a method that furthermore serves as an organizing framework for innumerable other applications and constructions. My preference, […]
  • Comment by Joel David Hamkins on Does every dense down-set in P(ω)/(fin) contain a partition?
    It is not a cBa, since no countably infinite antichain has a supremum, since by Hausdorff's method you can squeeze another set below any candidate. Every (ω,ω) gap is filled.
  • Comment by Joel David Hamkins on Does every dense down-set in P(ω)/(fin) contain a partition?
    This won't imply DC, since one can have AC for all sets at this level, and a violation of DC only very high up in the set-theoretic universe.
  • Comment by Joel David Hamkins on Extracting partitions from dense open subsets of complete Boolean algebras without choice
    Conceivably it is easier to find antichains and partitions in complete Boolean algebras than in partial orders. For example, every antichain in a cBa extends to a partition by taking the complement of the supremum. Do we know the question for partial orders is equivalent to the question for complete Boolean algebras?
  • Comment by Joel David Hamkins on Extracting partitions from dense open subsets of complete Boolean algebras without choice
    Calliope, you don't just want to say D≠{0}, but rather require a,b≠0 in your definition of density.
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    @user21820 I believe Caleb's point is that precisely because every Π10 statement is equivalent to a consistency statement, it is not actually so easy to have a robust conception of "ordinary" mathematics that excludes the kinds of issues that logicians investigate. Indeed, I find the category at bottom as meaningless and irritating as "natural". But […]
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    There is a whole literature on this.
  • Comment by Joel David Hamkins on Ordinary mathematics intrinsically requiring unbounded replacement/specification?
    I don't view that kind of justification as different in kind from the essentially similar story that people with the universe view tell about the cumulative hierarchy going all the way up in a totally deterministic manner, getting ZFC and much more, including large cardinals, on philosophical reflection grounds. In both cases, the justifications proceed […]

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